# Physics Cup – TalTech 2021 – Pr. 4, Hint 4.

Continuing with the previous hint, since the initial standard deviation of the speed of the electrons is much bigger than $2u$, once an electron has accumulated enough speed to overcome the barrier, its speed excess over the threshold is essentially a random value homogeneously distributed between 0 and $2u$. At this point, the phase diagram (with p-axis being vertical) is a curve consisting of an horizontally elongated rectangle (corresponding to the motion in the shock wave – the newly added section), joined with a vertically elongated rectangle (corresponding to the motion outside the shock wave), all together forming a 90-degrees-rotated-T-shaped pattern. During the further motion of the shock, this shape will change (eventually evolving into just a rectangle), but its surface area, the adiabatic invariant, is conserved. If $2u$ is small, the newly added rectangular section has a negligible surface area, which might not be the case if $2u$ is bigger. Based on all this you can deduce the speed distribution of the electrons at the end of the process, and hence, the pressure exerted by them.

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